Transport coefficients of graphene: Interplay of impurity scattering, Coulomb interaction, and optical phonons

We study the electric and thermal transport of the Dirac carriers in monolayer graphene using the Boltzmann-equation approach. Motivated by recent thermopower measurements [F. Ghahari, H.-Y.~Xie, T. Taniguchi, K. Watanabe, M.~S.~Foster, and P.~Kim, Phys.\ Rev.\ Lett.\ {\bf 116}, 136802 (2016)], we consider the effects of quenched disorder, Coulomb interactions, and electron--optical-phonon scattering. Via an unbiased numerical solution to the Boltzmann equation we calculate the electrical conductivity, thermopower, and electronic component of the thermal conductivity, and discuss the validity of Mott's formula and of the Wiedemann-Franz law. An analytical solution for the disorder-only case shows that screened Coulomb impurity scattering, although elastic, violates the Wiedemann-Franz law even at low temperature. For the combination of carrier-carrier Coulomb and short-ranged impurity scattering, we observe the crossover from the interaction-limited (hydrodynamic) regime to the disorder-limited (Fermi-liquid) regime. In the former, the thermopower and the thermal conductivity follow the results anticipated by the relativistic hydrodynamic theory. On the other hand, we find that optical phonons become nonnegligible at relatively low temperatures and that the induced electron thermopower violates Mott's formula. Combining all of these scattering mechanisms, we obtain the thermopower that quantitatively coincides with the experimental data.Comment: 20 pages, 9 figure

Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity

In this paper, we establish the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy on $\mathbb{R}^d$ in a low regularity Sobolev type space. More precisely, we reduce the regularity $s$ down to the currently known regularity requirement for unconditional uniqueness of solutions to the cubic nonlinear Schr\"odinger equation ($s\ge\frac{d}{6}$ if $d=1,2$ and $s>s_c=\frac{d-2}{2}$ if $d\ge 3$). In such a way, we extend the recent work of Chen-Hainzl-Pavlovi\'c-Seiringer.Comment: 26 pages, 1 figur

Mathematical Modeling of Product Rating: Sufficiency, Misbehavior and Aggregation Rules

Many web services like eBay, Tripadvisor, Epinions, etc, provide historical product ratings so that users can evaluate the quality of products. Product ratings are important since they affect how well a product will be adopted by the market. The challenge is that we only have {\em "partial information"} on these ratings: Each user provides ratings to only a "{\em small subset of products}". Under this partial information setting, we explore a number of fundamental questions: What is the "{\em minimum number of ratings}" a product needs so one can make a reliable evaluation of its quality? How users' {\em misbehavior} (such as {\em cheating}) in product rating may affect the evaluation result? To answer these questions, we present a formal mathematical model of product evaluation based on partial information. We derive theoretical bounds on the minimum number of ratings needed to produce a reliable indicator of a product's quality. We also extend our model to accommodate users' misbehavior in product rating. We carry out experiments using both synthetic and real-world data (from TripAdvisor, Amazon and eBay) to validate our model, and also show that using the "majority rating rule" to aggregate product ratings, it produces more reliable and robust product evaluation results than the "average rating rule".Comment: 33 page

Stochastic phenotype transition of a single cell in an intermediate region of gene-state switching

Multiple phenotypic states often arise in a single cell with different gene-expression states that undergo transcription regulation with positive feedback. Recent experiments have shown that at least in E. coli, the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed. Rather it is in the intermediate region which is difficult to handle mathematically.Under this condition, from a full chemical-master-equation description we derive a model in which the protein copy-number, for a given gene state, follow a deterministic mean-field description while the protein synthesis rates fluctuate due to stochastic gene-state switching. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers theory for chemical reactions. The resulting behaviors are significantly different from the two limiting cases studied previously.Comment: 6 pages,4 figure